I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just wondering, is there any big reason why the study of bundles would give information about varieties? (I suppose that for this matter, I should actually replace varieties by manifolds?)

I have heard about some invariants, like the Picard group for complex manifolds. But given my inexperience in these concepts, I don't really know why they should be important. So for those who are thinking about "cooking up invariants", some more detailed explanations on why they are useful (and hopefully some elementary examples!) would be appreciated. Thanks!

In the case of manifolds the primary reasons for studying vector bundles are the tangent and normal bundles, and the tubular neighbourhood theorem.
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Ryan BudneyDec 5 '09 at 6:23

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In algebraic geometry, to get the analogue of tubular neighborhoods, we need to study more general cones, in particular the normal cone, which in the smooth case reduces to the normal bundle.
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Charles SiegelDec 5 '09 at 18:20

7 Answers
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1) Subvarieties: Take a vector bundle, look at a section, where is it zero? Lots of subvarieties show up this way (not all, see this question) but generally, we can get lots of information out of vector bundles regarding subvarieties.

2) Invariants of spaces: The Picard group of Line bundles and more generally the Grothendieck group/ring is a useful invariant for differentiating spaces and analyzing the geometry indirectly. On smooth spaces, in fact, complexes of vector bundles can be used to replace coherent sheaves entirely (I believe by the Syzygy Theorem).

3) Maps into Projective Space: This one is line bundle specific. Let $V\to\mathbb{P}^n$ be any imbedding, say, then the pullback of $\mathcal{O}(1)$ is a line bundle on $V$. The nice thing is, the global sections of this line bundle determine and are determined by the map (we can get degenerate mappings by taking subspaces, but lets ignore that, and base loci for the moment). It turns out that we can define a line bundle to be ample, a condition just on the bundle, and that suffices to say that a power of it gives a morphism to $\mathbb{P}^n$, so understanding maps into projective space is the same thing as studying ample line bundles on a variety.

Hope that helps, there's a lot more, but those are the first three things that came to mind.

For (2), are you saying that the grothendieck group can be computed by looking at the locally free sheaves only? (I don't remember the exact conditions, but something like noetherian + regular + sth else)
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Ho Chung SiuDec 5 '09 at 5:15

I'm saying for smooth varieties, every coherent sheaf is quasi-isomorphic to a bounded complex of locally free sheaves, in fact, it's a resolution. So yeah, the Grothendieck group can be computed just from vector bundles. When things are more general than smooth varieties, I don't know. (I might even be assuming $k=\mathbb{C}$, I've lost track of which theorems I know use that, because I mostly work over $\mathbb{C}$)
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Charles SiegelDec 5 '09 at 5:18

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I believe you're using some version of the Hilbert syzygy theorem: the local ring of a smooth variety at any point has global dimension n (is this actually right?), so if you start with any sheaf, and do the first n-1 steps of the resolution (you might need projectivity to get the surjection from the vector bundle), then the nth is automatically a vector bundle, since its localization at every point is projective.
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Ben Webster♦Dec 6 '09 at 17:54

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Ah, after a little searching, I see that though it's morally like the Hilbert syzygy theorem, it's really a result of Serre that every regular local ring has finite global dimension equal to its Krull dimension (which is what the Hilbert syzygy theorem says about a polynomial ring).
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Ben Webster♦Dec 6 '09 at 18:05

I think many of the other answers boil down to the same underlying idea: Sections of vector bundles are "generalized functions" or "twisted functions" on your manifold/variety/whatever.

For example, Charles mentions subvarieties, which are roughly "zero loci of functions". However, there are no non-constant holomorphic global functions on, say, a projective variety. So how can we talk about subvarieties of a projective variety? Well, we do have non-constant holomorphic functions locally, so we can still define subvarieties locally as being zero loci of functions. But the functions $f_i$ which define a subvariety on one open set $U$ and the functions $g_i$ which define a subvariety on another open set $V$ won't necessarily agree on $U \cap V$. We need some kind of "twist" to make the $f_i$'s and the $g_i$'s match up on $U \cap V$. Upon doing so, the global object that we obtain is not a global function (because, again, there are no non-constant global functions) but a "twisted" global function, in other words a section of a vector bundle whose transition functions are described by these "twists".

Similarly, sections of vector bundles and line bundles are a nice way to talk about functions with poles. Meromorphic functions then become simply sections of a line bundle, which is nice because it allows us to avoid having to talk about $\infty$. This is essentially why line bundles are related to maps to projective space $X \to \mathbb{P}^n$; intuitively, $n+1$ sections of a line bundle over $X$ is the same as $n+1$ meromorphic functions on $X$, which is the same as a map "$X \to (\mathbb{C} \cup \infty)^{n+1}$" which becomes a map "$X \to \mathbb{P}^n$" after we "projectivize".

One way to think of vector bundles and their sections as being invariants of your manifold/variety/whatever is to think of them as describing what kinds of "generalized" or "twisted" functions are possible on your manifold/variety.

The view of sections of vector bundles as being "twisted functions" is also useful for physics, as in e.g. David's answer. For instance, suppose we have a manifold, which we think of as being some space in which particles are moving around. We have local coordinates on the manifold, which are used to describe the position of the particles. Since we are on a manifold, the transitions between the local coordinates are nontrivial. We may also be interested in studying, say, the velocities or momenta (or acceleration, etc.) of the particles moving around in space. On local charts we can describe these momenta easily in terms of the local coordinates, but then for a global description we need transitions between these local descriptions of momenta, just like how we need transitions between the local coordinates in order to describe the manifold globally. The transitions between local descriptions of momenta are not the same as that between the local coordinates (though the former depends on the latter); phrased differently, we obtain a non-trivial (ok, not always non-trivial, but usually non-trivial) vector bundle over our manifold.

this is the answer my advisor gave when i asked him why differential operators should act on sections of vector bundles. also the thom space construction explain the twisting well.
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Sean TilsonApr 2 '10 at 23:34

Although not a complete answer to the question, let me just point out that vector bundles are sometimes forced upon you.

For instance, you may start with an honest function $f$ defined on a manifold $M$ on which a group $G$ acts. Let's assume for simplicity that $G$ acts in such a way that the quotient $M/G$ is a manifold. If the function were invariant under the group, it would define an honest function on the quotient. But if the function is "almost" invariant, say
$$f(g^{-1} x) = \alpha(g) f(x)$$
for $g\in G$ and $x \in M$ and where $\alpha$ is some character of $G$, then $f$ only defines a section of a (homogeneous) line bundle on the quotient.

More generally if $f: M \to V$, where $\rho: G \to \mathrm{GL}(V)$ is a representation of $G$, and assuming that
$$f(g^{-1} x) = \rho(g) f(x)$$
then in the quotient $M/G$, $f$ defines a section of a (homogeneous) vector bundle.

Another case is where you have a family of endomorphism $\phi(x) \in \mathrm{End}(V)$ of a fixed vector space $V$, parametrised by a manifold $M$. Then the kernel of $\phi(x)$ is a vector subspace of $V$, and assuming that its dimension does not vary with $x$, define a vector bundle over $M$.

Also there are interesting invariants which require one to consider vector bundles. For instance, topological K-theory, which is the natural setting for the index theorem, is a theory of vector bundles.

Finally, vector bundles are essential for gauge theory which in turn have provided very useful results in topology: Donaldson's early work in the 80s on the topology of 4-manifolds, Seiberg-Witten theory in the mid 1990s,...

Another example where bundles are forced on you is if you want to differentiate a function on a manifold. (This may be a special case of one of the remarks above -- I write as a total non-expert.) If you differentiate a real-valued function on R^n you get a function that takes values in R^n: if you differentiate a real-valued function on an n-dimensional manifold then it takes values in the tangent bundle. This doesn't really explain why they are such a powerful concept, but at least it shows that they are a natural one.

I can accept that tangent bundles and vector fields, cotangent bundles and differential forms are natural constructions, but my problem is a bit different. An analogy: I can understand the study of a particular group representation could be useful, but (1) Why is the study of a general representation useful? (2) When we put all the representations altogether, sometimes it can reveal the information of the group. (e.g. \sum d_k^2 = n for finite groups) Why is that? (or in VB case, Picard group being an invariant of the manifold/variety)
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Ho Chung SiuDec 6 '09 at 3:28

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OK, in that case I think one has to turn to more sophisticated answers such as that you can use them to form K groups. If you'll excuse the indirect self-promotion, I'd recommend Burt Totaro's article on algebraic topology in the Princeton Companion to Mathematics, where he has quite a lot to say about bundles and why they are important.
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gowersDec 6 '09 at 11:11

Here are some motivations from the point of view geometric quantization. In this theory, one usually considers vector bundles over a symplectic manifold. The symplectic manifold represents the state space of a "classical mechanical system", for example in the case of a particle moving on a line, the symplectic manifold is R^2 the space of the particle's positions and momenta.

The equations of motion in classical mechanics (Hamilton's equations) are in general nonlinear in the state space coordinates, since they are obtained from Poisson brackets. The quantum version of this problem considers line bundles over the "classical" symplectic space. The sections of these bundles represent the particle's "wave functions". In quantum mechanics, the equations of motions are linear (The shrodinger equation). This "linearization" is achieved by working on the line bundle which has an intrinsically linear structure, and the induced evolution operators act linearly on the space of sections.

Many properties of line bundles have significance here also. For example, the projective space into which the sections of the line bundle define an embedding is just the projective quantum mechanical Hilbert space.

The generalization to a vector bundle is used to describe particles with "internal degrees of freedom" such as spin.

Harmonic analysis on vector bundles is used in spectrum problems of the quantum mechanical case.

Following Gowers' answer (which is what I would like to have written) and Siu's response, it seems to me that a common theme in mathematical research of the past, say, 50 years is something like the following:

2) Define the set or space of all such abstract objects and look for some kind of structure, usually algebraic or topological, that exists naturally on this space. If necessary, impose an equivalence condition (homeomorphic, homotopic, homologous)

3) By analyzing and classifying such spaces (which may depend on parameters), develop novel insights into the original examples that originally led to the abstract definition.

So, returning to vector bundles, once you have a theory for classifying and distinguishing different types of vector bundles, you can apply it to naturally defined bundles over manifolds or varieties and obtain new results on classifying and distinguishing different types of such spaces.

Vector bundles are particularly attractive, because they represent "linearizations" of the nonlinear structure of manifolds and varieties. So they are in many ways much easier to work with than the base spaces.

They are tractable and naturally occuring yet encode lots of information. They also provide a link between different mathematical techniques. One good comparison is between solving Yang Mills equations on a vector bundle and the Einstein equations on a Riemannian manifold.

K-Theory shows that there is lots of topological information contained in them.

Hitchin Kobayashi correspondence linking differential and algebraic techniques, the Atiyah Singer index theorem linking analysis and topology. Flat connections and curvature link geometry and representation theory. The Torelli theorem and Donaldson's work use them to reveal information about finer structures (algebraic and differentiable respectively). They occur naturally, tangent and normal bundles obviously but also projective embeddings.